Fixed Points of Automorphisms of Klein Surfaces 75
the form
(1) (g;±; [m1,...,mr]; {(n11,...,n1s1),...,(nk1,...,nksk)})
and mentioned above surface groups have signatures (g;±; [−]; {−}).
The numbers mi≥2 are called the proper periods, the brackets
(ni1,...,nisi), the period cycles, the numbers nij ≥2 are the link peri-
ods and g≥0 is said to be the orbit genus of Λ. The orbit space H/Λ
is a surface with kboundary components, orientable or not according
to the sign being + or −, having topological genus gand the canonical
projection H → H/Λ is a covering ramified over rinterior points with
ramification indices miand over sipoints lying on each boundary com-
ponent with ramification indices nij . A group with the signature (1) has
a presentation given by generators:
(a) xi,i= 1,...,r, (elliptic elements)
(b) cij ,i= 1, . . . , k, j = 0, . . . , si,(hyperbolic reflections)
(c) ei,i= 1,...,k, (boundary generators)
(d) ai, bi,i= 1,...,g if the sign is +,(hyperbolic translations)
di,i= 1,...,g if the sign is −,(glide reflections)
and relations
(1) xmi
i= 1, i=1,...,r,
(2) cisi=e−1
ici0ei,i=1,...,k,
(3) c2
ij = 1, i=1, . . . , k, j = 0,...,si,
(cij−1cij )nij = 1, i=1, . . . , k, j = 1,...,si,
(4) x1...xre1...eka1b1a
−1
1b
−1
1. . . agbga
−1
gb
−1
g=1 or
x1...xre1. . . ekd2
i...d2
g= 1.
Any system of generators of an NEC-group, which satisfies the above
relations, will be called a canonical system of generators and it is known,
that every element of finite order in Λ is conjugate either to a canonical
reflection or to a power of some canonical elliptic element xior else to a
power of the product of two consecutive canonical reflections cij−1,cij .
For every NEC-group we have an associated fundamental region,
whose hyperbolic area µ(Λ) depends only on the group and for a group
with signature (1) it is given by
(2) 2π εg +k−2 +
r
X
i=1
(1 −1/mi) + 1/2
k
X
i=1
si
X
i=1
(1 −1/nij)!,